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Abstract

The free oscillations of a functionally graded (FG) porous vertical cantilever beam in the frame work of Euler–Bernoulli beam theory is investigated. The beam is subjected to the gravity-load and the properties of the FG material such as the modulus of elasticity and the density are supposed to change through the thickness of the beam according to power-law relations. The equation of motion is derived using Newton’s second law. The Numerical Chebyshev collocation method is utilized to determine the transverse frequencies of the FG porous hanging and standing cantilever Euler–Bernoulli beams. A parametric study is conducted to determine the effects of various factors such as the transverse functionally graded index, the porosity factor, and the elastic and the mass density ratios on the natural frequencies and the mode shapes of the FG porous vertical hanging and standing cantilever thin beams under their self-weight. The accuracy of the proposed numerical method is evaluated through comparisons of the frequencies obtained from the present approach with those available in previous literature. In general, it was observed that the elastic ratio has a softening impact on the frequencies except for the fundamental frequency which remains constant as the elastic ratio increases. Moreover, the porosity parameter and the power-law index may have a softening or hardening impact on the frequencies, and the behavior of these frequencies depends on the values of the elastic and the mass density ratios.

Keywords

Natural vibrations functionally graded material spectral collocation method vertical cantilever beam porosity eigenvalue problem

Introduction

Due to their superior mechanical and thermal properties, Functionally Graded Materials (FGMs) have gained widespread attention and use in mechanical, aeronautical, and civil engineering.¹ The FGMs are a unique kind of composites where two or more constituent materials are combined to have properties that gradually vary through the dimensions of the structure. Moreover, porous materials are commonly utilized in several fields to manufacture light weight, stiffened, and energy-absorbing structures.^1,2 Besides, these materials are outstanding candidates for mitigating oscillations produced by wave propagation in sensitive applications such as the protection of electronic panels in spacecraft.^3,4 Therefore, the FG porous members have achieved the consideration of several scientists due to their excellent characteristics.⁵

It is known that vertical structures can be used to model many systems such as heavy columns, stiff rods, offshore structures, tall buildings, space shuttles, and water towers. As these structures are in a gravitational field, their self-weight will introduce a linearly varying axial tension, and as a result, they will undergo more oscillations.⁶

Many articles were devoted to investigating the dynamic behavior of structures with uniformly distributed axial load. For example, Paidoussis and Des Trois Maisons⁷ adopted the Galerkin approach to determine the fundamental natural frequency of a standing and hanging prismatic cantilever beam. Lakin⁸ utilized the method of matched asymptotic expansions to study the oscillations of a marginally stiff pendulum with a small bounce mass. Schafer⁹ employed the Ritz–Galerkin method, the series approximation, the regular perturbation theory, and the matched asymptotic expansions to determine analytical solutions for the eigenfrequencies of a vertical cantilever Euler–Bernoulli-beam. Yokoyama¹⁰ examined the dynamic behavior of uniform hanging Timoshenko beams under gravity. The finite difference approach was employed and the impacts of the self-weight, shear distortion, and rotary inertia on the vibration behavior. Naguleswaran¹¹ carried out the natural vibrations of a vertical uniform cantilever beam under the self-weight effect. The standing and hanging cases were considered and a solution that involves the superposition method was applied to derive the frequency equations.

Moreover, Lee¹² conducted the stability analysis of a rotating vertical cantilever beam. Hamilton’s principle was utilized to formulate the equation of motion. The assumed mode and Bolotin’s methods were used to determine the regions of instability. Wang et al.¹³ investigated the primary and secondary resonance of hanging cantilever beam under the condition of three-to-one internal resonance caused by gravity. The performance was discussed numerically and analytically. Additionally, Bizzi et al.¹⁴ used the “Hankel-Airy” relations to obtain the analytical solutions for the natural vibrations of a Rayleigh–Timoshenko beam under its self-weight. Virgin et al.¹⁵ examined the influence of self-weight on the oscillations of vertical cantilever beams, where it was shown that in the hanging case the beam will be stiffened by the beam’s weight, whereas for the standing orientation, the beam will undergo a de-stiffening impact. Wang¹⁶ explored the stability and free vibrations of a standing non-uniform cantilever beam, where an initial value numerical technique was applied to compute the first three frequencies. It was shown that the properties of the cross section and the taper have remarkable influence on the frequencies.

Li et al.¹⁷ presented an integral equation approach to develop expressions for the fundamental frequencies of vertical cantilever Rayleigh beams with a tip mass where the result of gravity on the frequency is specified explicitly. Hijmissen and van Horssen¹⁸ exploited a method of multiple time-scales to obtain the natural frequencies of a vertical Timoshenko beam. Abramovich¹⁹ studied the natural vibrations of hanging composite thick beams in the framework of Timoshenko beam hypothesis. The impacts of gravity, shear distortion, and rotary inertia on the natural frequencies of the hanging beam were examined.

Based on the above literature review, it is noticed that studies with specific emphasis on the natural oscillations of functionally graded porous hanging and standing cantilever Euler–Bernoulli beams are scarce. Hence, the purpose of this study is to fill this gap in literature. Therefore, the main contributions of this study consist of the utilization of the spectral collocation technique to analyze the free vibration characteristics of FG porous hanging and standing cantilever beams. The rest of this research is arranged as follows. In Section 2, the numerical Chebyshev Spectral Collocation Method (CSCM) is introduced. In Section 3, the equation of motion is presented, and the CSCM is utilized to formulate the eigenvalue problem. The suggested method is verified, and the findings are displayed and examined in Section 4. In Section 5, the conclusions are stated and further developments are suggested.

Chebyshev Spectral Collocation Method (CSCM)

As shown in Figure 1, the Chebyshev points are given on the range of [-1, 1] as²⁰

x_{j} = \cos (\frac{j π}{N}), j = 0, 1, \dots ., N

(1)

Figure 1.

The Chebyshev points.

The elements of the Chebyshev differentiation matrix, ${[D]}_{N}$ are expressed as

{(D_{N})}_{00} = \frac{2 N^{2} + 1}{6}, {(D_{N})}_{N N} = - \frac{2 N^{2} + 1}{6}, {(D_{N})}_{j j} = \frac{- x_{j}}{2 (1 - x_{j}^{2})}, j = 1, \dots, N - 1 {(D_{N})}_{i j} = \frac{c_{i}}{c_{j}} \frac{{(- 1)}^{i + j}}{(x_{i} - x_{j})}, i \neq j, i, j = 0, \dots, N . c_{i} = {\begin{cases} 2, & i = 0 o r N, \\ 1, & o t h e r w i s e \end{cases}

(2)

The nth derivative of a function is defined as $D_{n} = {(D_{N})}^{n}$ . In the present article, the spatial coordinate is shifted to the range of [0, 1], hence, the elements of $D_{N}$ will be adjusted.

Theory and problem formulation

Mathematical model

A functionally graded porous cantilever bean with length L, thickness h, and width b is shown in Figure 2. In this article, the modulus of elasticity $E (z)$ and the density of the material $ρ (z)$ alter through the thickness of the cantilever beam and are given as²¹

E (z) = ((E_{2} - E_{1}) {(\frac{2 (z + d) + h}{2 h})}^{n} + E_{1} - \frac{α}{2} (E_{2} + E_{1}))

(3)

ρ (z) = ((ρ_{2} - ρ_{1}) {(\frac{2 (z + d) + h}{2 h})}^{n} + ρ_{1} - \frac{α}{2} (ρ_{2} + ρ_{1}))

(4)where z is the horizontal axis taken at distance d from the middle surface (

z_{m s}

) of the cantilever beam,

α

is the porosity parameter,

n

is the power-law index. Subscript 1 is for the left surface, whereas subscript 2 refers to the right surface. The distance d is defined as²²

d = \frac{(E_{r} - 1) n h}{2 (2 + n) (E_{r} + n - (\frac{α}{2} (n + 1) (1 + E_{r})))}, E_{r} = \frac{E_{2}}{E_{1}}

(5)

Figure 2.

The functionally graded porous gravity-loaded cantilever (a) hanging and (b) standing beam.

Following Naguleswaran,¹¹ the bending moment $M (x, t)$ is given by

M (x, t) = E (z) I \frac{\partial^{2} w}{\partial x^{2}}

(6)where

I

is the area moment of inertia about the y axis (out of plane), and

E (z) I

is the flexural rigidity of the beam and is given as

E I = \int_{A} ((E_{2} - E_{1}) {(\frac{2 (z + d) + h}{2 h})}^{n} + E_{1} - \frac{α}{2} (E_{2} + E_{1})) z^{2} d A

(7)The shear force

Q (x, t)

is defined as

Q (x, t) = - \frac{\partial M}{\partial x} + T (x) \frac{\partial w}{\partial x}

(8)where the axial tension

T (x)

is given as¹¹

T (x) = ρ (z) A g (L - x)

(9)where

A

is the area of the beam’s cross section, and the mass per unit length

ρ (z) A

is given as

ρ A = \int_{A} ((ρ_{2} - ρ_{1}) {(\frac{2 (z + d) + h}{2 h})}^{n} + ρ_{1} - \frac{α}{2} (ρ_{2} + ρ_{1})) d A

(10)

Following Naguleswaran, the summation of forces on a beam element in the z-axis yields¹¹

\frac{\partial Q}{\partial x} = ρ (z) A \frac{\partial^{2} w}{\partial t^{2}}

(11)

Substituting equations (4), (6), and (7) into equation (9), yields

ρ A \frac{\partial^{2} w}{\partial t^{2}} + E I \frac{\partial^{4} w}{\partial x^{4}} + ρ A g (x - L) \frac{\partial^{2} w}{\partial x^{2}} + ρ A g \frac{\partial w}{\partial x} = 0

(12)For convenience, the following non-dimensional parameters are introduced as

X = \frac{x}{L}, \hat{W} = \frac{w}{L}, Z = \frac{z}{h}, d = \frac{\hat{d}}{h}, T = t \sqrt{\frac{E_{1} I}{ρ_{1} A L^{4}}}, E_{r} = \frac{E_{2}}{E_{1}}, ρ_{r} = \frac{ρ_{2}}{ρ_{1}}, γ = \frac{ρ_{1} A g L^{3}}{E_{1} I}, Ω = ω \sqrt{\frac{ρ_{1} A L^{4}}{E_{1} I}}

(13)

In view of the non-dimensional variables, the governing equation of motion is obtained as

{\bar{I}}_{0} \frac{\partial^{2} \hat{W}}{\partial T^{2}} + {\bar{D}}_{11} \frac{\partial^{4} \hat{W}}{\partial X^{4}} + {\bar{I}}_{0} γ (X - 1) \frac{\partial^{2} \hat{W}}{\partial X^{2}} + {\bar{I}}_{0} γ \frac{\partial \hat{W}}{\partial X} = 0

(14)where

{\bar{I}}_{0} = \int_{- d - 0.5}^{0.5 - d} {(ρ_{r} - 1) {(z + d + 0.5)}^{n} + 1 - \frac{α}{2} (ρ_{r} + 1)} d z

(15)

{\bar{D}}_{11} = 12 \int_{- d - 0.5}^{0.5 - d} {(E_{r} - 1) {(z + d + 0.5)}^{n} + 1 - \frac{α}{2} (E_{r} + 1)} z^{2} d z

(16)

For harmonic free vibrations, the separation of variables is used, and the transverse displacement is given as²³

\hat{W} (X, T) = W (X) e^{i Ω T}

(17)

Accordingly, equation (12) can be given as

{\bar{D}}_{11} \frac{d^{4} W}{d X^{4}} + {\bar{I}}_{0} γ (X - 1) \frac{d^{2} W}{d X^{2}} + {\bar{I}}_{0} γ \frac{d W}{d X} = {{\bar{I}}_{0} Ω}^{2} W

(18)

The boundary conditions are given as¹¹

Clamped end

w (x, t) = 0 and \frac{\partial w (x, t)}{\partial x} = 0

(19)

Free end

Q (x, t) = 0 and M (x, t) = 0

(20)

Solution procedure

The CSCM is employed to discretize equations (18)–(20). For a hanging cantilever beam, the boundary conditions are given as

W_{1} = 0

D 1 {(1, 1) W}_{1} + D 1 {(1, 2) W}_{2} + D 1 {(1, N) W}_{N} + D 1 {(1, N + 1) W}_{N + 1} = - \sum_{k = 3}^{N - 1} D 1 (1, k) W_{k}

D 2 {(N + 1, 1) W}_{1} + D 2 {(N + 1, 2) W}_{2} + D 2 {(N + 1, N) W}_{N} + D 2 {(N + 1, N + 1) W}_{N + 1} = - \sum_{k = 3}^{N - 1} D 2 (N + 1, k) W_{k}

D 3 {(N + 1, 1) W}_{1} + D 3 {(N + 1, 2) W}_{2} + D 3 {(N + 1, N) W}_{N} + D 3 {(N + 1, N + 1) W}_{N + 1} = - \sum_{k = 3}^{N - 1} D 3 (N + 1, k) W_{k}

(21)

Equation (21) can be expressed as

[\begin{array}{l} D 1 (1, 2) & D 1 (1, N) & D 1 (1, N + 1) \\ D 2 (N + 1, 2) & D 2 (N + 1, N) & D 2 (N + 1, N + 1) \\ D 3 (N + 1, 2) & D 3 (N + 1, N) & D 3 (N + 1, N + 1) \end{array}] {\begin{cases} W_{2} \\ W_{N} \\ W_{N + 1} \end{cases}} = {\begin{cases} - \sum_{k = 3}^{N - 1} D 1 (1, k) W_{k} \\ - \sum_{k = 3}^{N - 1} D 2 (N + 1, k) W_{k} \\ - \sum_{k = 3}^{N - 1} D 3 (N + 1, k) W_{k} \end{cases}}

(22)

The displacements $W_{2}$ , $W_{N}$ , and $W_{N + 1}$ are given as

{\begin{cases} W_{2} \\ W_{N} \\ W_{N + 1} \end{cases}} = {[\begin{array}{l} D 1 (1, 2) & D 1 (1, N) & D 1 (1, N + 1) \\ D 2 (N + 1, 2) & D 2 (N + 1, N) & D 2 (N + 1, N + 1) \\ D 3 (N + 1, 2) & D 3 (N + 1, N) & D 3 (N + 1, N + 1) \end{array}]}^{- 1} {\begin{cases} - \sum_{k = 3}^{N - 1} D 1 (1, k) W_{k} \\ - \sum_{k = 3}^{N - 1} D 2 (N + 1, k) W_{k} \\ - \sum_{k = 3}^{N - 1} D 3 (N + 1, k) W_{k} \end{cases}}

(23)

Equation (23) is rewritten as

W_{2} = - A_{1, 1} \sum_{k = 3}^{N - 1} D 1 (1, k) W_{k} - A_{1, 2} \sum_{k = 3}^{N - 1} D 2 (N + 1, k) W_{k} - A_{1, 3} \sum_{k = 3}^{N - 1} D 3 (N + 1, k) W_{k},

(24)

W_{N} = - A_{2, 1} \sum_{k = 3}^{N - 1} D 1 (1, k) W_{k} - A_{2, 2} \sum_{k = 3}^{N - 1} D 2 (N + 1, k) W_{k} - A_{2, 3} \sum_{k = 3}^{N - 1} D 3 (N + 1, k) W_{k},

(25)

W_{N + 1} = - A_{3, 1} \sum_{k = 3}^{N - 1} D 1 (1, k) W_{k} - A_{3, 2} \sum_{k = 3}^{N - 1} D 2 (N + 1, k) W_{k} - A_{3, 3} \sum_{k = 3}^{N - 1} D 3 (N + 1, k) W_{k},

(26)where

{[\begin{array}{l} D 1 (1, 2) & D 1 (1, N) & D 1 (1, N + 1) \\ D 2 (N + 1, 2) & D 2 (N + 1, N) & D 2 (N + 1, N + 1) \\ D 3 (N + 1, 2) & D 3 (N + 1, N) & D 3 (N + 1, N + 1) \end{array}]}^{- 1} = [\begin{array}{l} A_{1, 1} & A_{1, 2} & A_{1, 3} \\ A_{2, 1} & A_{2, 2} & A_{2, 3} \\ A_{3, 1} & A_{3, 2} & A_{3, 3} \end{array}]

(27)Thus, for a hanging cantilever beam, the Dn matrix is adjusted as

\bar{D n} (i, k) = \sum_{k = 3}^{N - 1} D n (i, k) - D n (i, 2) (A_{1, 1} \sum_{k = 3}^{N - 1} D 1 (1, k) + A_{1, 2} \sum_{k = 3}^{N - 1} D 2 (N + 1, k) + A_{1, 3} \sum_{k = 3}^{N - 1} D 3 (N + 1, k))

- D n (i, N) (A_{2, 1} \sum_{k = 3}^{N - 1} D 1 (1, k) + A_{2, 2} \sum_{k = 3}^{N - 1} D 2 (N + 1, k) + A_{2, 3} \sum_{k = 3}^{N - 1} D 3 (N + 1, k))

- D n (i, N + 1) (A_{3, 1} \sum_{k = 3}^{N - 1} D 1 (1, k) + A_{3, 2} \sum_{k = 3}^{N - 1} D 2 (N + 1, k) + A_{3, 3} \sum_{k = 3}^{N - 1} D 3 (N + 1, k))

(28)where

i = 3, 4 …, N-1.

In a similar manner, the boundary conditions for a standing cantilever are presented as

W_{N + 1} = 0

D 1 {(N + 1, 1) W}_{1} + D 1 {(N + 1, 2) W}_{2} + D 1 {(N + 1, N) W}_{N} = - \sum_{k = 3}^{N - 1} D 1 (N + 1, k) W_{k}

D 2 {(1, 1) W}_{1} + D 2 {(1, 2) W}_{2} + D 2 {(1, N) W}_{N} = - \sum_{k = 3}^{N - 1} D 2 (1, k) W_{k}

- β D 3 {(1, 1) W}_{1} + γ D 1 {(1, 1) W}_{1} - β D 3 {(1, 2) W}_{2} + γ D 1 {(1, 2) W}_{2} - β D 3 {(1, N) W}_{N} + γ D 1 {(1, N) W}_{N} = - \sum_{k = 3}^{N - 1} (- β D 3 {(1, k) W}_{k} + γ D {(1, k) W}_{k})

(29)where

β = \frac{{\bar{D}}_{11}}{{\bar{I}}_{0}}

(30)Equation (29) is rewritten in a more compact form as

Q {(1, 1) W}_{1} + Q {(1, 2) W}_{2} + Q {(1, N) W}_{N} = - \sum_{k = 3}^{N - 1} Q (1, k) W_{k}

(31)where

Q (1, k) = - β D 3 {(1, k) W}_{k} + γ D 1 {(1, k) W}_{k}

(32)

Equation (32) is expressed in matrix-vector form as

[\begin{array}{l} D 2 (1, 1) & D 2 (1, 2) & D 2 (1, N) \\ Q (1, 1) & Q (1, 2) & Q (1, N) \\ D 1 (N + 1, 1) & D 1 (N + 1, 2) & D 1 (N + 1, N) \end{array}] {\begin{cases} W_{1} \\ W_{2} \\ W_{N} \end{cases}} = {\begin{cases} - \sum_{k = 3}^{N - 1} D 2 (1, k) W_{k} \\ - \sum_{k = 3}^{N - 1} Q (1, k) W_{k} \\ - \sum_{k = 3}^{N - 1} D 1 (N + 1, k) W_{k} \end{cases}}

(33)

The displacements $W_{1}$ , $W_{2}$ , and $W_{N}$ are given as

{\begin{cases} W_{1} \\ W_{2} \\ W_{N} \end{cases}} = {[\begin{array}{l} D 2 (1, 1) & D 2 (1, 2) & D 2 (1, N) \\ Q (1, 1) & Q (1, 2) & Q (1, N) \\ D 1 (N + 1, 1) & D 1 (N + 1, 2) & D 1 (N + 1, N) \end{array}]}^{- 1} {\begin{cases} - \sum_{k = 3}^{N - 1} D 2 (1, k) W_{k} \\ - \sum_{k = 3}^{N - 1} Q (1, k) W_{k} \\ - \sum_{k = 3}^{N - 1} D 1 (N + 1, k) W_{k} \end{cases}}

(34)

The displacements $W_{1}$ , $W_{2}$ , and $W_{N}$ are rewritten as

W_{1} = - B_{1, 1} \sum_{k = 3}^{N - 1} D 2 (1, k) W_{k} - B_{1, 2} \sum_{k = 3}^{N - 1} Q (1, k) W_{k} - B_{1, 3} \sum_{k = 3}^{N - 1} D 1 (N + 1, k) W_{k},

(35)

W_{2} = - B_{2, 1} \sum_{k = 3}^{N - 1} D 2 (1, k) W_{k} - B_{2, 2} \sum_{k = 3}^{N - 1} Q (1, k) W_{k} - B_{2, 3} \sum_{k = 3}^{N - 1} D 1 (N + 1, k) W_{k},

(36)

W_{N} = - B_{3, 1} \sum_{k = 3}^{N - 1} D 2 (1, k) W_{k} - B_{3, 2} \sum_{k = 3}^{N - 1} Q (1, k) W_{k} - B_{3, 3} \sum_{k = 3}^{N - 1} D 1 (N + 1, k) W_{k},

(37)where

{[\begin{array}{l} D 2 (1, 1) & D 2 (1, 2) & D 2 (1, N) \\ Q (1, 1) & Q (1, 2) & Q (1, N) \\ D 1 (N + 1, 1) & D 1 (N + 1, 2) & D 1 (N + 1, N) \end{array}]}^{- 1} = [\begin{array}{l} B_{1, 1} & B_{1, 2} & B_{1, 3} \\ B_{2, 1} & B_{2, 2} & B_{2, 3} \\ B_{3, 1} & B_{3, 2} & B_{3, 3} \end{array}]

(38)For a standing cantilever beam, the (Dn) matrix is revised as

\bar{D n} (i, k) = \sum_{k = 3}^{N - 1} D n (i, k) - D n (i, 1) (B_{1, 1} \sum_{k = 3}^{N - 1} D 2 (1, k) + B_{1, 2} \sum_{k = 3}^{N - 1} Q (1, k) + B_{1, 3} \sum_{k = 3}^{N - 1} D 1 (N + 1, k))

- D n (i, 2) (B_{2, 1} \sum_{k = 3}^{N - 1} D 2 (1, k) + B_{2, 2} \sum_{k = 3}^{N - 1} Q (1, k) + B_{2, 3} \sum_{k = 3}^{N - 1} D 1 (N + 1, k))

- D n (i, N) (B_{3, 1} \sum_{k = 3}^{N - 1} D 2 (1, k) + B_{3, 2} \sum_{k = 3}^{N - 1} Q (1, k) + B_{3, 3} \sum_{k = 3}^{N - 1} D 1 (N + 1, k))

(39)where

i = 3, 4 \dots, N - 1 .

The equation of motion for a FG porous gravity-loaded cantilever beam is given using the presented method as

(β \bar{D 4} + γ Λ \bar{D 2} + γ \bar{D 1}) W = Ω^{2} W

(40)where

Λ

is a diagonal (N-3)

\times

(N -3) matrix with the entries

x_{i} - 1

. The transverse dimensionless natural frequencies and the corresponding mode shapes of the FG porous cantilever beam are obtained by finding the eigenvalues and the eigenvectors of equation (40).

Results and discussion

To ensure the accuracy of the proposed solution approach, the first three transverse frequencies of isotropic cantilever vertical beams are determined and compared with those obtained by Naguleswaran¹¹ as shown in Table 1. The comparison shows that present solutions are in exact agreement with Naguleswaran¹¹ results, which validates the efficiency of the present solution technique.

Table 1.

The first three frequencies of hanging and standing cantilever beams.

		Hanging			Standing
$γ$		$Ω_{1}$	$Ω_{2}$	$Ω_{3}$	$Ω_{1}$	$Ω_{2}$	$Ω_{3}$
1000	Naguleswaran¹¹	38.69551	92.28009	160.70198	27.39606	106.09241	188.31185
1000	Present study	38.69551	92.28009	160.70198	27.39606	106.09241	188.31185
500	Naguleswaran¹¹	27.58902	67.61250	123.47107	20.93861	80.59374	146.26870
500	Present study	27.58902	67.61250	123.47107	20.93861	80.59374	146.26870
100	Naguleswaran¹¹	12.86874	36.50917	78.98033	11.77221	46.07428	91.28511
100	Present study	12.86874	36.50917	78.98033	11.77221	46.07428	91.28511
50	Naguleswaran¹¹	9.46853	30.21719	70.97195	9.440162	37.70100	79.00464
50	Present study	9.46853	30.21719	70.97195	9.440162	37.70100	79.00464
10	Naguleswaran¹¹	5.29178	23.91200	63.68155	5.98253	26.65655	65.76094
10	Present study	5.29178	23.91200	63.68155	5.98253	26.65655	65.76094
0	Naguleswaran¹¹	3.51602	22.03449	61.69721	3.516015	22.03449	61.69721
0	Present study	3.51602	22.03449	61.69721	3.516015	22.03449	61.69721

The variations of the first four dimensionless frequencies of a hanging cantilever FG porous beam versus the elastic ratio $E_{r}$ at several values of the transverse FG index $n$ are illustrated in Figures 3 (a)–(c). It is observed that elastic ratio $E_{r}$ has a softening impact on the frequencies except for the first frequency that remains constant as $E_{r}$ increases. Moreover, it is depicted that the higher modes are more sensitive to the change in the elastic ratio $E_{r}$ when compared to the lower modes. Furthermore, these plots show that the frequencies decrease as the FG index n grows.

Figure 3.

The variations of the first four dimensionless frequencies of a hanging cantilever Euler–Bernoulli FG porous beam with the elastic ratio $E_{r}$ for $= 0.1$ , $ρ_{r} = 2$ , and $γ = 200$ at (a) $n = 0.1$ (b) $n = 1.0$ (c) $n = 3.0$ .

The influence of the porosity parameter $α$ on the dimensionless frequencies of a standing cantilever FG porous beam with $E_{r} = 5$ , $ρ_{r} = 0.5$ , and $γ = 50$ is presented in Figure 4. For n = 0.1, $α$ has a strengthening influence on the frequencies. On the contrary, for n = 1.0 and 2.5, the frequencies decline as the porosity parameter propagates. The higher modes are more sensitive to the variation in the porosity parameter when compared to the lower ones. It is worth mentioning that for n = 2.5, the fundamental frequency reaches zero when $α = 0.4$ . Thus, one may conclude that a standing FG porous cantilever FG beam with $E_{r} = 5$ , $ρ_{r} = 0.5$ , and $γ = 50$ is unstable for $α > 0.4$ . As observed from Figure 4, the results indicate that the frequencies drop as the FG index n grows.

Figure 4.

The first four dimensionless frequencies of a Standing cantilever Euler–Bernoulli FG porous beam as functions of the porosity parameter for $E_{r} = 5$ , $ρ_{r} = 0.5$ , $γ = 50$ at (a) $n = 0.1$ (b) $n = 1.0$ (c) $n = 2.5$ .

Figure 5 illustrates the effects of the porosity factor on the first four dimensionless transverse frequencies of a standing cantilever Euler–Bernoulli FG porous beam with $E_{r} = 0.5$ , $ρ_{r} = 5$ , and $γ = 50$ . As depicted from Figures 5(a) and 5(b) for the FG index n = 0.1 and 1.0, respectively, the first four dimensionless frequencies get smaller as $α$ increases, whereas as presented in Figure 5(c) for the FG index n = 3.0, the transverse frequencies rise as the porosity factor grows. It is obvious that the frequencies are less sensitive to the variation in $α$ for n = 1.0 in comparison with those for n = 0.1 and 3.0. Moreover, it is noticed that the frequencies of a standing cantilever FG porous beam with $E_{r} = 0.5$ , $ρ_{r} = 5$ , and $γ = 50$ increase as the FG index n advances.

Figure 5.

The first four dimensionless frequencies of a Standing cantilever Euler–Bernoulli FG porous beam as functions of the porosity parameter for $E_{r} = 0.5$ , $ρ_{r} = 5$ , $γ = 50$ at (a) $n = 0.1$ (b) $n = 1.0$ (c) $n = 3$ .

The variations of the first four dimensionless frequencies of a standing cantilever FG porous beam with $E_{r} = 5$ , $ρ_{r} = 0.25$ , and $γ = 50$ versus the porosity parameter $α$ are displayed in Figure 6. For n = 0.1, the frequencies rise as the porosity parameter $α$ advances. On the other hand, it is observed that the porosity factor has a diminishing influence on the transverse frequencies for n = 1.0 and 2.0. Additionally, the frequencies decline as the FG index n grows.

Figure 6.

The influence of the porosity factor $α$ on the first four dimensionless transverse frequencies of a standing cantilever FG porous beam with $E_{r} = 5$ , $ρ_{r} = 0.25$ , and $γ = 50$ (a) n = 0.1 (b) n = 1.0 (c) n = 2.0.

In Figures 7 (a)–(c), the dimensionless frequencies of a standing Euler–Bernoulli cantilever FG porous beam with $E_{r} = 0.25$ , $ρ_{r} = 5.0$ , and $γ = 300$ are plotted versus the porosity parameter $α$ . The frequencies decrease as the porosity parameter rises for n = 0.1 and n = 1.0. However, the porosity parameter has a hardening impact on the frequencies for n = 6.0. For the parameters given in these plots, the transverse frequencies rise as the FG index n propagates. These plots show that the influence of the porosity factor and the FG index on the frequencies is non-monotonic.

Figure 7.

The Alterations of the first four dimensionless transverse frequencies of a standing Euler–Bernoulli FG porous cantilever beam with the porosity parameter $α$ with $E_{r} = 0.25$ , $ρ_{r} = 5.0$ , and $γ = 300$ (a) n = 0.1 (b) n = 1.0 (c) n = 6.0.

The impact of the FG index n on the dimensionless transverse frequencies of a hanging Euler–Bernoulli cantilever FG porous beam with $E_{r} = 2.0$ , $α = 0.1$ , and $γ = 200$ at different values of the mass density ratio $ρ_{r}$ is illustrated in Figure 8. The plots reveal that for $ρ_{r}$ = 0.5 and 1.0, the frequencies get smaller as the FG index n progresses. In contrast, for $ρ_{r}$ = 5.0, the frequencies grow as the FG index n develops. It is worthwhile to mention that the fundamental frequencies are insensitive to the change in the FG index n regardless the value of the mass density ratio $ρ_{r}$ . Moreover, it is observed that the mass density ratio $ρ_{r}$ has a softening influence on the frequencies.

Figure 8.

The influence of the FG index n on the $first four dimensionless frequencies$ of a hanging Euler–Bernoulli cantilever FG porous beam with $E_{r} = 2.0$ , $α = 0.1$ , and $γ = 200$ (a) $ρ_{r}$ = 0.5 (b) $ρ_{r}$ = 1.0 (c) $ρ_{r}$ = 5.0.

In Figures 9(a)–(c), the first four frequencies of a of a hanging cantilever FG porous beam with $ρ_{r} = 4.0$ , $α = 0.1$ , and $γ = 200$ are plotted versus the FG index n. The plots reveal that for $E_{r}$ = 0.5 and $E_{r}$ = 1.0, the frequencies propagate as n advances except for the fundamental frequencies, where the FG index n has a negligible impact on the frequencies. In Figure 9(c), except for the fundamental frequency, it can be deduced that the frequencies drop as n alters over the range 0 to 1.0, and then they rise as n increases. In fact, the trend of the variation of the natural frequencies with respect to the FG index may not be monotonic.

Figure 9.

The impact of the FG index n on the dimensionless transverse frequencies of a hanging Euler–Bernoulli cantilever FG porous beam with $ρ_{r} = 4.0$ , $α = 0.1$ , and $γ = 200$ (a) $E_{r}$ = 0.5 (b) $E_{r}$ = 1.0 (c) $E_{r}$ = 5.0.

Figures 10 (a)–(b) shows the first two mode shapes of a hanging Euler–Bernoulli cantilever FG porous beam with $ρ_{r} = 5.0$ , $E_{r} = 2.0$ , n = 1, and $α = 0.1$ at various values of the gravity factor $γ$ . Figure 9(a) illustrates that as $γ$ gets larger, the displacements are slightly increased for X $ϵ$ [0, 0.7], while $γ$ has almost no impact on the first mode shape for X $ϵ$ [0.7, 1.0]. In Figure 10(b), it is noticed that as $γ$ advances, the displacements decline for X $ϵ$ [0.2, 0.65], whereas $γ$ has a marginal impact on the deflections for $X < 0.2$ and $X > 0.65$ . The third and fourth mode shapes for the beam considered in Figure 2(a) are presented in Figures 11(a)–(b), where it is shown that the gravity parameter $γ$ has a remarkable influence on these mode shapes.

Figure 10.

The first two mode shapes for a hanging Euler–Bernoulli cantilever FG porous beam with $ρ_{r} = 5.0$ , $E_{r} = 2.0$ , n = 1, and $α = 0.1$ .

Figure 11.

The third and fourth mode shapes for a hanging Euler–Bernoulli cantilever FG porous beam with $ρ_{r} = 5.0$ , $E_{r} = 2.0$ , n = 1, and $α = 0.1$ .

In Figures 12(a)–(b), the first two mode shapes of a standing Euler–Bernoulli cantilever FG porous beam with $ρ_{r} = 5.0$ , $E_{r} = 0.25$ , n = 0.1, and $γ = 500$ at several values of the porosity parameter $α$ . It is observed from Figure 12(a) that the increase in $α$ has a negligible influence on the first mode shape, whereas the second mode shape is significantly affected for $α = 0.4$ as shown in Figure 12(b). The third and fourth mode shapes for the beam considered in Figure 2(b) are plotted in Figures 13(a)–(b).

Figure 12.

The first two mode shapes of a standing Euler–Bernoulli cantilever FG porous beam with $ρ_{r} = 5.0$ , $E_{r} = 0.25$ , n = 0.1, and $γ = 500$ .

Figure 13.

The third and fourth mode shapes for a standing Euler–Bernoulli cantilever FG porous beam with $ρ_{r} = 5.0$ , $E_{r} = 0.25$ , n = 0.1, and $γ = 500$ .

Conclusions

In this article, the free oscillations of FG porous hanging and standing cantilevers were analyzed. The modulus of elasticity and the density of the material are graded through the thickness of the beam following a power-law relation. The mathematical model was obtained using Newton’s second law of motion, and the numerical CSCM was applied to convert the differential governing equation of motion and the corresponding boundary conditions into algebraic relations. The impacts of the transverse FG index, the porosity factor, the elastic and mass density ratios and the gravity parameter on the transverse dimensionless frequencies and mode shapes were examined. It is believed that the results and the proposed approach may be useful for the researchers investigating the dynamic behavior of FG porous members. The main conclusions are:

• The elastic ratio has a softening impact on the frequencies except for the fundamental frequency which remains constant as the elastic ratio increases.

• Depending on the values of the elastic and the mass density ratios, the porosity parameter and the power-law index may have a softening or hardening impact on the frequencies.

• For the cases considered in the current study, the mass density ratio has a softening impact on the frequencies.

• For hanging and standing cantilevers, it was found that the higher mode shapes are more sensitive to the change in the gravity parameter than the lower ones.

For the future work, it is suggested to carry out the free and forced vibrations of FG porous Timoshenko and Rayleigh cantilever beams, and to investigate the dynamic behavior of two-dimensional structures such as plates and shells. Additionally, it is suggested to perform the natural and forced dynamic behavior of uniform and tapered FG porous structures.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ma’en S Sari

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Free vibrations of functionally graded porous hanging and standing cantilever beams

Abstract

Keywords

Introduction

Chebyshev Spectral Collocation Method (CSCM)

Theory and problem formulation

Mathematical model

Solution procedure

Results and discussion

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References